> It is claimed that S = Q^2 \/ (R\Q)^2 is path connected.> How can the be?

I accidentally introduced the space S up in a May 2002 sci.maththread that you participated in, and I believe in one ofyour posts in that thread you came up with the limitingzig-zag line segment method. ("Accidentally, because youbegan the thread asking about another dense subset of R^2,and I wasn't paying close enough attention to realize thatthe space you originally brought up was not the space myfirst post in that thread discussed.)

By the way, does anyone know of a reference to the pathwiseconnectedness of S (defined above) that predates Fall 1987?The earliest "reference" I know of is my own, when I provedthat S is pathwise connected in a topology class (the actualproblem assigned was to prove S is connected, but I managedto get pathwise connected).

For more about this, see the posts cited in the followingAugust 2008 sci.math post of mine.

is pathwise connected in R^2 (the plane), where Q is theset (subspace of R, actually) of rational numbers and P isthe set of irrational numbers. For more about this, and howsomeone besides me wound up publishing it, see the first postbelow. Incidentally, I misread the problem being consideredby the original poster in that thread, thinking it was theproblem I had solved in 1987, but it was something different.However, I think the rest of the post is fine, except forone tangential comment I made in order to work in a Star Warspun, which is taken care of in the 16 May 2002 post. The lastpost below includes an update where I found a simpler proofof the pathwise connectedness of Q^2 union P^2 in a recenttopology text. [In all, you can find 3 proofs in these poststhat this space is pathwise connected, and at least one proofindependent of these 3 proofs that this space is connected.]